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The sub-campaign to define Mitt Romney’s tenure at Bain Capital, as I never tire of pointing out, is merely about softening up the Republican nominee for the major fight of the campaign: Obama’s charge that his economic program is merely a retread of the Bush-era program of tax cuts for the rich. Over several months, Romney has laid the groundwork for his own defense. He has promised that his tax plan will not cut taxes for the rich (below the levels established under Bush). Recall that Romney’s old campaign line about how he wasn’t concerned about the very poor was also packaged with a supposedly parallel line about not being concerned about the very rich — neither group would receive any particular targeted benefit from his program.

Romney’s plan has been to hold together these promises by shrouding his tax and budget plans in a veil of secrecy. Romney has promised to reduce tax rates across the board by 20 percent, which would offer huge tax cuts for the rich. But he has promised to close unspecified deductions in the tax code so as to offset the cost, and leave the rich paying the same effective tax rate. Indeed, Romney has boasted about his strategy, noting that its lack of detail means it “can’t be scored,” and thus Obama can’t prove that his plan really would cut taxes for the rich.

But oh yes, he can.

The Brookings Institution and the Tax Policy Center today released a study of Romney’s proposals, insofar as they are known. The finding was simple. Romney’s promises, it found, are mathematically impossible. The amount of revenue available from tax deductions for the rich is smaller than the amount of revenue lost by cutting tax rates for the rich. Even if Romney sincerely scoured the tax code and wiped out every last tax break for the rich that he hasn’t promised to preserve (he has promised to keep in place tax incentives for saving, like the capital gains tax break), the rich will pay lower rates and a lower share of the tax burden.

It’s worth noting that the study embraces implausibly friendly assumptions as to how Romney would go about this. It assumes he would ruthlessly purge the tax code of breaks for the rich, even highly popular ones like the charitable deduction. It further assumes that, in order to wring every last penny out of the rich, Romney would cut off all deductions immediately for every dollar in income over $200,000 a year. (In reality, nobody would create a tax code that meant going from $200,000 a year to $200,001 would jack up your taxes by thousands of dollars — you would ramp up the tax deduction phase-in, which would reduce taxes for the rich even more. But the paper bends over backwards to grant Romney this implausible assumption.)

What’s more, the paper assumes that Romney’s plan would increase economic growth, meaning it wouldn’t have to find dollar-for-dollar replacements for all its lost income. To measure this cheerful scenario, the paper adopts a model created by Greg Mankiw — who is, of course, a Bush administration veteran and one of Romney’s main economic advisers.

Piling implausibly optimistic assumption upon implausibly optimistic assumption, the paper nonetheless concludes that Romney will cut taxes for the rich. That means it would result in some combination of higher taxes for the middle class or higher deficits. If you take Romney at his word that he would hold tax revenue steady at its current levels, then he would be implementing a significant shift in the tax burden from the rich to the middle class. 95% of all taxpayers would pay more taxes, in order to finance a tax cut for the most affluent.

And remember, this is assuming the most favorable possible case for Romney. Under more realistic assumptions — that he won’t close every single penny in tax deductions benefitting the rich, and that his plan won’t spur economic growth to the degree a Republican like Mankiw hopes it would — then the transfer from the non-rich to the rich would be even higher. All of which shows why, despite the constant drumbeat of conservative pleas for him to unveil more policy specifics, Romney is going to keep his proposals as vague as possible.